Related papers: An Efficient Generalized Shift-Rule for the Prefer…
A nonbinary Ford sequence is a de Bruijn sequence generated by simple rules that determine the priorities of what symbols are to be tried first, given an initial word of size $n$ which is the order of the sequence being generated. This set…
Classic cycle-joining techniques have found widespread application in creating universal cycles for a diverse range of combinatorial objects, such as shorthand permutations, weak orders, orientable sequences, and various subsets of $k$-ary…
Let $S$ be a string of length $n$ with characters from an alphabet of size $\sigma$. The \emph{subsequence automaton} of $S$ (often called the \emph{directed acyclic subsequence graph}) is the minimal deterministic finite automaton…
We generalize the notion of a de Bruijn sequence to a "multi de Bruijn sequence": a cyclic or linear sequence that contains every k-mer over an alphabet of size q exactly m times. For example, over the binary alphabet {0,1}, the cyclic…
We present sufficient conditions for when an ordering of universal cycles $\alpha_1, \alpha_2, \ldots, \alpha_m$ for disjoint sets $\mathbf{S}_1, \mathbf{S}_2, \ldots , \mathbf{S}_m$ can be concatenated together to obtain a universal cycle…
A universal cycle for a set S of combinatorial objects is a cyclic sequence of length |S| that contains a representative of each element in S exactly once as a substring. Despite the many universal cycle constructions known in the…
A de Bruijn covering code is a q-ary string S so that every q-ary string is at most R symbol changes from some n-word appearing consecutively in S. We introduce these codes and prove that they can have length close to the smallest possible…
A subsequence of a word $w$ is a word $u$ such that $u = w[i_1] w[i_2] , \dots w[i_{|u|}]$, for some set of indices $1 \leq i_1 < i_2 < \dots < i_k \leq |w|$. A word $w$ is $k$-subsequence universal over an alphabet $\Sigma$ if every word…
A de Bruijn sequence of order $k$ over a finite alphabet is a cyclic sequence with the property that it contains every possible $k$-sequence as a substring exactly once. Orthogonal de Bruijn sequences are collections of de Bruijn sequences…
The watchman's walk problem in a digraph calls for finding a minimum length closed dominating walk, where direction of arcs is respected. The watchman's walk of a de Bruijn graph of order $k$ is described by a de Bruijn sequence of order…
The move structure represents a permutation $\pi$ of $[0,n)$ as a covering set of $O(r)$ disjoint intervals (contiguous subsets of $[0,n)$), where $r$ is the minimum number of intervals whose values permute together. Formally, $r = 1 +…
We introduce the notion of unavoidable (complete) sets of word patterns, which is a refinement for that of words, and study certain numerical characteristics for unavoidable sets of patterns. In some cases we employ the graph of pattern…
A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy\in E$. Word-representable graphs generalize several important classes of graphs such…
Common intervals have been defined as a modelisation of gene clusters in genomes represented either as permutations or as sequences. Whereas optimal algorithms for finding common intervals in permutations exist even for an arbitrary number…
Given a word $w$ and a Parikh vector $\mathcal{P}$, an abelian run of period $\mathcal{P}$ in $w$ is a maximal occurrence of a substring of $w$ having abelian period $\mathcal{P}$. Our main result is an online algorithm that, given a word…
A de Bruijn cycle is a cyclic listing of length A, of a collection of A combinatorial objects, so that each object appears exactly once as a set of consecutive elements in the cycle. In this paper, we show the power of de Bruijn's original…
We propose a general greedy algorithm for binary de Bruijn sequences, called Generalized Prefer-Opposite (GPO) Algorithm, and its modifications. By identifying specific feedback functions and initial states, we demonstrate that most…
We define generalized de Bruijn words as those words having a Burrows-Wheeler transform that is a concatenation of permutations of the alphabet. We show that generalized de Bruijn words are in 1-to-1 correspondence with Hamiltonian cycles…
A balanced generalized de Bruijn sequence with parameters $(n,l,k)$ is a cyclic sequence of $n$ bits such that (a) the number of 0's equals the number of 1's, and (b) each substring of length $l$ occurs at most $k$ times. We determine…
We develop a new approach to random walks on de Bruijn graphs over the alphabet $A$ through right congruences on $A^k$, defined using the natural right action of $A^+$. A major role is played by special right congruences, which correspond…